The IIT Foundation series of Mathematics Class

Trishna Knowledge Systems

Chapter 17 Modular Arithmetic - all with Video Answers

Educators

Chapter Questions

02:25

Problem 1

In the set of integers modulo $8,28 \otimes_{8} 2=$
(1) 0
(2) 1
(3) 2
(4) 3

Kumareshwaran Rathinasabapathy

Numerade Educator

01:08

Problem 2

If $25 \equiv 4(\bmod p)$, where $p$ is a prime number, then $p$ is
(1) 3
(2) 5
(3) 7
(4) either (1) or (3)

Kumar Vaibhav

Numerade Educator

01:22

Problem 3

Solve for $x$, if $5 x \equiv 0(\bmod 4)$
(1) 0
(2) 3
(3) 2
(4) Both (1) and (2).

Kumar Vaibhav

Numerade Educator

01:56

Problem 4

In the set of integers modulo $12,38 \oplus_{12} 28=$
(1) 6
(2) 5
(3) 4
(4) 3

Kumareshwaran Rathinasabapathy

Numerade Educator

01:34

Problem 5

The largest single digit number that satisfies $14 \mathrm{x} \equiv 4(\bmod 3)$ is
(1) 5
(2) 7
(3) 8
(4) 9

Kumar Vaibhav

Numerade Educator

01:03

Problem 6

If 8th August of 2009 is a Saturday, then 15 August of 2010 falls on
(1) Saturday
(2) Sunday
(3) Wednesday
(4) Thursday

Kumar Vaibhav

Numerade Educator

01:04

Problem 7

If $23 \equiv 7(\bmod x)$, then which of the following cannot be the value of $x$ ?
(1) 4
(2) 6
(3) 8
(4) 16

Kumar Vaibhav

Numerade Educator

02:12

Problem 8

Find the remainder when $13^{15}$ is divided by 5 .
(1) 4
(2) 3
(3) 2
(4) 1

Kumar Vaibhav

Numerade Educator

00:46

Now the time is $1: 30 \mathrm{p}, \mathrm{m}$. If I woke up 8 hours ago, then I woke up at
(1) 4: $30 \mathrm{a} \cdot \mathrm{m}$.
(2) $5: 30 \mathrm{a}, \mathrm{m}$.
(3) $3: 30 \mathrm{a} \cdot \mathrm{m}$.
(4) $6: 30 \mathrm{a} \cdot \mathrm{m}$.

Kumar Vaibhav

Numerade Educator

01:00

Problem 10

If $37 \equiv 18(\bmod p)$, where $p$ is a prime number, then find $p$.
(1) $\underline{3}$
(2) 7
(3) 19
(4) Either (1) or (2).

Kumar Vaibhav

Numerade Educator

01:16

Problem 11

If $15 \equiv 3(\bmod x)$, then which of the following cannot be the value of $x$ ?
(1) 3
(2) 4
(3) 6
(4) 8

Kumareshwaran Rathinasabapathy

Numerade Educator

03:40

Problem 12

Find the remainder when $5^{18}$ is divided by $19 .$
(1) 1
(2) 4
(3) 11
(4) 17

Kumar Vaibhav

Numerade Educator

04:27

Problem 13

The largest two-digit number that satisfies $5 x \equiv 6(\bmod 4)$ is
(1) 96
(2) 97
(3) 98
(4) 99

Kumareshwaran Rathinasabapathy

Numerade Educator

01:07

Problem 14

If you were born on 15 April 1993 which was a Tuesday, then on which day of the week did your birthday fall in 1994 ?
(1) Tuesday
(2) Wednesday
(3) Thursday
(4) Monday

Kumar Vaibhav

Numerade Educator

02:00

Problem 15

Find the remainder when $11^{12}$ is divided by 7 .
(1) 0
(2) 1
(3) 3
(4) 5

Kumar Vaibhav

Numerade Educator

01:26

Problem 16

If $a \equiv b(\bmod m)$ and the remainder obtained when $^{4}$ ' is divided by $m$ is 2 , then find the remainder when 'b' is divided by $\mathrm{m}$.
(1) 2
(2) 1
(3) 0
(4) Cannot be determined

Kumareshwaran Rathinasabapathy

Numerade Educator

00:59

Problem 17

If $x \equiv y(\bmod 2)$, then which of the following is/are correct?
(a) $x$ is even and $y$ is odd
(b) Both $\mathrm{x}$ and $\mathrm{y}$ are odd.
(c) Both $\mathrm{x}$ and $\mathrm{y}$ are even.
(1) Only (c)
(2) Only (a)
(3) Both
(b) and (c)
(4) Both (a) and (b)

Kumar Vaibhav

Numerade Educator

00:59

Problem 18

If 1st January, 2010 is a Friday, then the fifth Sunday of January, 2011 will fall on
(1) 26 th day
(2) 27 th day
(3) 29 th day
(4) 30 th day

Kumar Vaibhav

Numerade Educator

01:31

Problem 19

Anand started a work on Sunday at $9: 30 \mathrm{a.m}$. and he finished it after 87 hours there of Then he finished the work on
(1) wednesday 11: 30 p.m.
(2) thursday $0: 30$ a.m.
(3) wednesday 0: 30 a.m.
(4) thursday 11: 30 p.m.

Kumar Vaibhav

Numerade Educator

01:31

Problem 20

Which of the following are the common solutions of $3 x \equiv 0(\bmod 6)$ and $2 x \equiv 0(\bmod 4) ?$
(a) 0
(b) 2
(c) 4
(1) Both (a) and (b)
(2) Both (a) and (c)
(3) Both (b) and (c)
(4) All of (a), (b) and (c)

Kumar Vaibhav

Numerade Educator

01:22

Problem 21

If $15 x \equiv 2(\bmod 3)$, then which of the following is a possible value of $x ?$
(1) 3
(2) 315
(3) 0
(4) None of these

Kumar Vaibhav

Numerade Educator

01:31

Problem 22

Which of the following is a common solution for $6 \mathrm{x} \equiv 0(\bmod 8)$ and $8 \mathrm{x} \equiv 0(\bmod 10) ?$
(1) 0
(2) 4
(3) 6
(4) Both (1) and (2)

Kumar Vaibhav

Numerade Educator

03:58

Problem 23

Find the remainder when $2^{24}$ is divided by 35 .
(1) 2
(2) 31
(3) 1
(4) 29

Kumareshwaran Rathinasabapathy

Numerade Educator

05:36

Problem 24

Which of the following is correct?
(1) $5 \oplus_{3} 2 \equiv 3 \otimes_{3} 6(\bmod 4)$
(2) $4 \oplus_{3} 2 \equiv 3 \otimes_{4} 5(\bmod 6)$
(3) $5 \oplus_{5} 3 \equiv 6 \otimes_{8} 9(\bmod 3)$
(4) None of these

Kumareshwaran Rathinasabapathy

Numerade Educator

08:22

Problem 25

Anas Venkitta

Numerade Educator

01:37

Problem 26

If $x$ belongs to the set of residues modulo 10 then the common solution of $5+x \equiv 0(\bmod 3)$ and 6 $+\mathrm{x} \equiv 0(\bmod 5)$ is
(1) 1
(2) 2
(3) 4
(4) 5

Kumar Vaibhav

Numerade Educator

01:05

Problem 27

By which of the following numbers should $3^{5}$ be divided to obtain a remainder 3?
(1) 7
(2) 11
(3) 5
(4) None of these

Kumar Vaibhav

Numerade Educator

01:34

Problem 28

Find the remainder when $6^{11}-6$ is divided by 11 .
(1) 5
(2) 1
(3) 0
(4) None of these

Kumareshwaran Rathinasabapathy

Numerade Educator

01:06

Problem 29

Find $x$, if $9 x \equiv 2(\bmod 7)$
(1) 1
(2) 2
(3) 3
(4) 4

Kumar Vaibhav

Numerade Educator

03:45

Problem 30

Find the remainder when $3^{19}$ is divided by $19 .$
(1) 3
(2) 15
(3) 16
(4) None of these

Kumareshwaran Rathinasabapathy

Numerade Educator

03:14

Problem 31

In the set of integers modulo $9,15 \otimes_{9} 10=$
(1) 3
(2) 6
(3) 0
(4) 1

Kumareshwaran Rathinasabapathy

Numerade Educator

01:01

Problem 32

If $7 x \equiv 1(\bmod 5)$, then which of the following is a possible value of $x$ ?
(1) 10
(2) 11
(3) 12
(4) None of these

Kumar Vaibhav

Numerade Educator

02:41

Problem 33

In the set of integers modulo $17,19 \oplus_{17} 15=$
(1) 0
(2) 1
(3) 2
(4) 3

Kumareshwaran Rathinasabapathy

Numerade Educator

00:39

Problem 34

Sangeeta started singing on Monday at $10.30 \mathrm{a} . \mathrm{m}$. in order to enter her name in the Guinness Book of world records. If she sings continuously for 36 hours, then she will finish her singing on
(1) Tuesday $10.30 \mathrm{a}, \mathrm{m}$.
(2) Wednesday $10.30 \mathrm{a}, \mathrm{m}$.
(3) Tuesday $10.30$ p.m.
(4) Wednesday $10.30$ p.m.

Kumar Vaibhav

Numerade Educator

02:03

Problem 35

Which of the following is a common solution of $3 \mathrm{x} \equiv 2(\bmod 5)$ and $4 \mathrm{x} \equiv 0(\bmod 6) ?$
(1) 9
(2) 4
(3) 6
(4) None of these

Kumar Vaibhav

Numerade Educator

00:43

Problem 36

Find the remainder when $3^{215}$ is divided by $43 .$
(1) 35
(2) 28
(3) 33
(4) 30

Nida Wasiq

Numerade Educator

01:53

Problem 37

Kishore reached his school on Monday at $8: 30 \mathrm{a} \cdot \mathrm{m}$. and then immediately started on a tour to GOA. After $1061 / 2$ hours there on, he reached his house. Then Kishore reached his house on
(1) Saturday at 7 p.m.
(2) Friday at 6 p.m.
(3) Saturday at 6 p.m.
(4) Friday at 7 p.m.

Kumar Vaibhav

Numerade Educator

02:10

Problem 38

If 1 st August, 2012 is Wednesday, then find the day on which we shall celebrate our Independence Day in the year 2015 .
(1) Saturday
(2) Sunday
(3) Friday
(4) Thursday
Find the remainder when $5^{97}$ is divided by $97 .$

Kumar Vaibhav

Numerade Educator

00:54

Problem 39

Find the remainder when $5^{9 \%}$ is divided by $97 .$
(1) 5
(2) 97
(3) 92
(4) None of these

Kumareshwaran Rathinasabapathy

Numerade Educator

03:12

Problem 40

If $a \equiv b(\bmod m)$, then which of the following is not always true?
(1) $a^{2} \equiv b^{2}(\bmod m)$
(2) $a+m \equiv b+m(\bmod 2 m)$
(3) $a m \equiv b m\left(\bmod m^{2}\right)$
(4) None of these

Kumareshwaran Rathinasabapathy

Numerade Educator

00:57

Problem 41

If $x^{3} \equiv x(\bmod 3)$, then $x$ can be
(1) 2
(2) 5
(3) 4
(4) All of these

Kumar Vaibhav

Numerade Educator

02:29

Problem 43

A part of Caley's table for $\otimes_{6}$ is given below. Find the values of $x, y$ and $z$.
$$
\begin{array}{|c|c|c|c|c|c|}
\hline \otimes_{6} & 0 & 1 & 2 & 3 & 4 \\
\hline 0 & 0 & 0 & 0 & 0 & 0 \\
\hline 1 & 0 & 1 & 2 & 3 & 4 \\
\hline 2 & 0 & 2 & 4 & 0 & 2 \\
\hline 3 & 0 & 3 & 0 & y & 0 \\
\hline 4 & 0 & 4 & z & 0 & x \\
\hline
\end{array}
$$
(1) $x=4, y=3, z=2$
(2) $x=2, y=4, z=3$
(3) $x=3, y=4, z=2$
(4) $\mathrm{x}=4, \mathrm{y}=2, \mathrm{z}=3$

Kumareshwaran Rathinasabapathy

Numerade Educator

02:51

Problem 43

A part of Caley's table for $\oplus_{7}$ is given below. Find the value of $\mathrm{p}, \mathrm{q}, \mathrm{x}$ and $\mathrm{y} .$
\begin{tabular}{|c|c|c|c|c|}
\hline$\oplus_{7}$ & 1 & 2 & 3 & 4 \\
\hline 1 & 2 & 3 & 4 & 5 \\
\hline 2 & 3 & 4 & 5 & 6 \\
\hline 3 & 4 & 5 & $\mathrm{p}$ & $\mathrm{x}$ \\
\hline 4 & 5 & $\mathrm{Q}$ & $\mathrm{y}$ & 1 \\
\hline
\end{tabular}
(1) $x=y=0, p=q=5$
(2) $x=y=1, p=q=6$
(3) $x=y=3, p=q=4$
(4) $x=y=0, p=q=6$

Kumareshwaran Rathinasabapathy

Numerade Educator

01:12

Problem 44

The Independence day of India in 2007 was celebrated on a Wednesday, then Children's day in 2008 was celebrated on a
(1) Friday
(2) Saturday
(3) Sunday
(4) Monday

Nida Wasiq

Numerade Educator

00:40

Problem 45

If $13 \equiv 3($ mod $p)$, then $p$ can be
(1) 2
(2) 5
(3) 10
(4) All of these

Kumar Vaibhav

Numerade Educator

00:49

Problem 46

If the 1st January of a certain year, which was not a leap year, was a Thursday, then what day of the week was the be 31 st December of that year?
(1) Monday
(2) Thursday
(3) Sunday
(4) Saturday

Kumar Vaibhav

Numerade Educator

01:04

Problem 47

If $x+10 \equiv 1(\bmod 8)$, then $x$ can be
(1) 1
(2) 0
(3) 6
(4) 7

Kumar Vaibhav

Numerade Educator

00:48

Problem 48

If $\mathrm{x}$ belongs to the set of residues modulo 6 and $3+\mathrm{x} \equiv 2(\bmod 6)$, then $\mathrm{x}=$
(1) 1
(2) 3
(3) 4
(4) 5

Kumar Vaibhav

Numerade Educator

01:07

Problem 49

The 2nd January, 2009 is a Friday. The fourth Sunday of January 2010 falls on the
(1) 2 3rd day
(2) 24 th day
(3) 25 th day
(4) 26 th day

Kumar Vaibhav

Numerade Educator

04:39

Problem 50

Which of the following is/are correct?
(1) $5 \oplus_{2} 4 \equiv 17 \otimes_{5} 3(\bmod 7)$
(2) $6 \oplus_{4} 7 \equiv 19 \otimes_{0} 3(\bmod 3)$
(3) $9 \oplus_{7} 3 \equiv 8 \otimes_{7} 9(\bmod 9)$
(4) None of these

Anas Venkitta

Numerade Educator

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